Surface Area of Simple 3D Shapes 07:22 minutes

Video Transcript

Transcript Surface Area of Simple 3D Shapes

On what should be a happy night in Wonderland the queen is throwing one of her famous tantrums. She just got a new throne delivered and it's NOT red! The queen summons the Mad Hatter to fix the problem immediately. How much paint will the Mad Hatter need to cover the throne? To answer this question, we'll need to calculate the surface area of simple 3D shapes. Surface area is the sum of the areas of all the surfaces of a shape. The queen's throne is a COMPOSITE of SEVERAL shapes. What shapes do you see? We have a rectangular prism here a triangular prism here and four cylindrical legs. Let's look at the rectangular prism first. This prism is 25 22 4 inches. The top and bottom faces are rectangles with dimensions 25 by 22. So the area of one of these faces is length times width, or 550 square inches. That makes their combined area twice that, or 1100 square inches. The FRONT face is 22 by 4 so the area is 88 square inches. The BACK face is identical so that's 176 square inches for both of these sides combined. The LEFT face is 25 by 4 so the area is 100 square inches. With the identical right face, that makes 200 square inches combined. Adding all these areas together gives us a surface area of 1,476 square inches. In general, we can find the surface area of a rectangular prism with this formula 2 times length times width, plus 2 times width times height, plus 2 times length times height. The majestic back of the throne is a TRIANGULAR prism. The FRONT and BACK faces of this shape are ISOSCELES TRIANGLES, with LEGS of 39 inches and BASE of 30 inches. This triangle has a HEIGHT of 36 inches. The area of a TRIANGLE is one-half of the base times the height. That's half of 30 times 36 which is 15 times 36, or 540. Because there is an identical triangle on the BACK...we double that to get 1,080 square inches. Now to calculate the LATERAL AREA of this shape, or the area of all the remaining faces combined. To do that, imagine placing the prism so that one triangular face is on the ground. Now, unfolding the sides gives us one long rectangle, with a height of 3 inches. The length of this long rectangle is the same as the PERIMETER of our triangle, or the length of all three sides combined. The AREA of this long rectangle can be found by multiplying the length, 108 inches, times the height, 3 inches...which gives us 324 square inches. We add the area of the triangles and the area of the lateral faces to get a surface area of 1,404 square inches. In general, we can find the surface area of any prism in two steps. First, find the area of the BASE of the prism and double it. Then, find the PERIMETER of the base and multiply it times the height to get the LATERAL area of all the sides combined. Add those together and we've got the surface area of the prism. We can use this idea to find the surface area of the four cylinders that compose the legs of the queen's chair, as well. The height of each cylinder is 15 inches. The circular bases of each cylinder have a radius of 2 inches. So what does that make the area of one circular base? Using the area formula for a circle pi times the radius 2 squared gives us an area of 4 pi for each base. We double that to get a combined area of 8 pi for the top and bottom. Now, how would you calculate the area of the lateral face? If we unroll the lateral face, we can see that it's a rectangle with a height of 15 inches. But what's the length? The length of the lateral face is the same as the CIRCUMFERENCE of the circular BASE. Do you remember how to find the circumference of a circle? That's 2 pi times the radius, 2, or 4 pi. So the lateral area is 4 pi times 15, or 60 pi. Adding the faces together, that gives us a surface area of 68 pi per cylinder. Now, to find the total surface area of the throne, we add all our parts together. 1,476 square inches for the rectangular prism plus 1,404 square inches for the triangular prism plus 4 times the surface area of one cylindrical prism, since we have four legs. Using 3.14 as an approximation for pi that gives us a TOTAL surface area of about 3,734.08 square inches. To review, we can find the surface area of prisms and cylinders in just a few steps. First, find the area of the BASE, and double it. That might mean finding the area of a rectangle a triangle or a circle. Then, find the perimeter or circumference of the base. Multiply the perimeter or circumference times the height to find the lateral area. Add the area of the bases and the lateral area to get the total surface area. By calculating the total surface area, the Mad Hatter was able to use just the right amount of paint on the throne. It looks perfect, but wait a second, Your Highness! Oh no... Now the queen is definitely seeing red.